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In a small, plain room at the Boston Children’s Museum, scientists are asking deep questions about the foundations of human knowledge — with the help of toddlers and squeaky balls. At the Museum of Science, they are watching children play, gaining insights into how teaching works.

Among the lessons: Teaching too much can stifle exploration.

Until recently, mystery surrounded the precise ways in which babies and toddlers begin to make sense of the world. But researchers, with clever experiments at these museums and elsewhere, are finding that young children have a surprisingly sophisticated intuitive grasp of probabilities, which they use to make inferences. When a toy does not work, or a squeeze ball squeaks, even babies weigh data and make informed bets about why.

The results are forming the basis for a new understanding of one of the most distinctive traits of the human mind — the ability to make, test, and continually adjust ideas about how one thing causes another. Such insights could help classroom teachers.

“We start with these newborn babies and by the time they’re 4 years old, they have a lot of common sense knowledge about the world. They have ideas about physics, they have ideas about other people, they have ideas about causal relationships,’’ said Laura Schulz, associate professor of cognitive sciences at the Massachusetts Institute of Technology. “The way we get the world right is by making bets — based on probabilities, given the evidence.’’

Children make different types of bets, however, depending on how they are taught. In a new study published in the journal Cognition, Schulz and colleagues built a complicated toy and brought it to the Museum of Science. The toy had many functions — pulling on one tube caused a squeak, pressing a button turned on a light, and pushing a pad made music play.

In one scenario, an adult instructed preschoolers in how the toy worked, demonstrating only the squeak sound. In another, children were just shown the toy.

In both cases, the researchers encouraged the children to play with the toy and figure out how it worked. Those who were explicitly taught how to make the toy squeak played with it for less time and discovered fewer functions than those who did not receive a tutorial.

How children learned that the toy squeaked also affected their behavior. When the adult seemed to accidentally trigger the squeak instead of explicitly showing them how to use the toy, children explored more and discovered more aspects of the toy than when they received instructions.

The experiments suggest that teaching must strike a balance: Obviously, children need to be able to effectively draw on existing knowledge in order to function and move forward. But teaching can also limit children’s impulse to investigate and test their own ideas, and how something is taught to a child can affect his or her exploration.

“People on the front lines feel as if there’s a tremendous pressure to make the environment for young children more and more academic — less and less exploratory,’’ said Alison Gopnik, a professor of psychology at the University of California, Berkeley, and author of “The Philosophical Baby.’’ “Even something that looks like random, exploratory play can help children to learn and in some cases help them to learn better.’’

What has become clear is that even when they are not being formally instructed, babies are picking up on all kinds of cues that affect how they learn. Schulz’s work suggests that well before they are introduced to the notion of math, babies are weighing probabilities.

Hyowon Gweon, a graduate student at MIT who works with Schulz, regularly visits a playroom at the Children’s Museum, asking parents if they would allow their young toddlers to participate in an experiment that could illuminate how the developing mind works.

In a recent study, she presented more than 100 15-month-olds with a box with a clear front, so they could see the balls inside. For some of the children, the box contained mostly yellow balls, along with a few blue ones. For others, the box was filled with mostly blue balls.

In all cases, Gweon drew three blue balls from the box, squeezing each one to produce a squeak. Then, she drew a fourth, yellow ball and placed it in front of the babies to see what they would do.

Toddlers shown a box filled mostly with yellow balls were less likely to try squeezing the yellow ball — seeming to conclude that the researcher must have carefully shown them the blue balls because they squeak. But toddlers whose box was mostly full of blue balls tended to try squeezing the yellow ball — apparently guessing that the researcher had picked the blue balls by chance.

Gweon tested other scenarios as well. She shook a box of mostly yellow balls, allowing three blue balls to fall out — each of which she squeaked — followed by a yellow. Again, the toddlers seemed to take the seemingly random ball selection into account; they tended to squeeze the yellow ball. Well before they’ve developed the ability to count the balls or calculate a probability, they seemed to be weighing the likelihood a ball would be purposefully or randomly drawn from the box.

Psychology researchers have agreements with the museums, allowing them to approach parents and ask whether their children will participate in experiments. At the Children’s Museum, the experiments take place in a small “PlayLab’’ just steps away from an indoor playground.

The experiments are brief — timed to the minutes-long attention span of young children. At the end, each child gets a certificate “for successful completion of a course in causal learning at the Massachusetts Institute of Technology’’ — and a sticker that says, “I helped!’’